# What are State Space Models? — State Space Models: A Modern Approach

# What are State Space Models?

    # meta-data does not work yet in VScode
    # https://github.com/microsoft/vscode-jupyter/issues/1121

    {
        "tags": [
            "hide-cell"
        ]
    }

    ### Install necessary libraries

    try:
        import jax
    except:
        # For cuda version, see https://github.com/google/jax#installation
        %pip install --upgrade "jax[cpu]"
        import jax

    try:
        import jsl
    except:
        %pip install git+https://github.com/probml/jsl
        import jsl

    try:
        import rich
    except:
        %pip install rich
        import rich

    {
        "tags": [
            "hide-cell"
        ]
    }

    ### Import standard libraries

    import abc
    from dataclasses import dataclass
    import functools
    import itertools

    from typing import Any, Callable, NamedTuple, Optional, Union, Tuple

    import matplotlib.pyplot as plt
    import numpy as np

    import jax
    import jax.numpy as jnp
    from jax import lax, vmap, jit, grad
    from jax.scipy.special import logit
    from jax.nn import softmax
    from functools import partial
    from jax.random import PRNGKey, split

    import inspect
    import inspect as py_inspect
    import rich

    from rich import inspect as r_inspect
    from rich import print as r_print

    def print_source(fname):
        r_print(py_inspect.getsource(fname))

\[ \begin{align}\begin{aligned}\newcommand\floor[1]{\lfloor#1\rfloor}\\\newcommand{\real}{\mathbb{R}}\\% Numbers \newcommand{\vzero}{\boldsymbol{0}} \newcommand{\vone}{\boldsymbol{1}}\\% Greek https://www.latex-tutorial.com/symbols/greek-alphabet/ \newcommand{\valpha}{\boldsymbol{\alpha}} \newcommand{\vbeta}{\boldsymbol{\beta}} \newcommand{\vchi}{\boldsymbol{\chi}} \newcommand{\vdelta}{\boldsymbol{\delta}} \newcommand{\vDelta}{\boldsymbol{\Delta}} \newcommand{\vepsilon}{\boldsymbol{\epsilon}} \newcommand{\vzeta}{\boldsymbol{\zeta}} \newcommand{\vXi}{\boldsymbol{\Xi}} \newcommand{\vell}{\boldsymbol{\ell}} \newcommand{\veta}{\boldsymbol{\eta}} %\newcommand{\vEta}{\boldsymbol{\Eta}} \newcommand{\vgamma}{\boldsymbol{\gamma}} \newcommand{\vGamma}{\boldsymbol{\Gamma}} \newcommand{\vmu}{\boldsymbol{\mu}} \newcommand{\vmut}{\boldsymbol{\tilde{\mu}}} \newcommand{\vnu}{\boldsymbol{\nu}} \newcommand{\vkappa}{\boldsymbol{\kappa}} \newcommand{\vlambda}{\boldsymbol{\lambda}} \newcommand{\vLambda}{\boldsymbol{\Lambda}} \newcommand{\vLambdaBar}{\overline{\vLambda}} %\newcommand{\vnu}{\boldsymbol{\nu}} \newcommand{\vomega}{\boldsymbol{\omega}} \newcommand{\vOmega}{\boldsymbol{\Omega}} \newcommand{\vphi}{\boldsymbol{\phi}} \newcommand{\vvarphi}{\boldsymbol{\varphi}} \newcommand{\vPhi}{\boldsymbol{\Phi}} \newcommand{\vpi}{\boldsymbol{\pi}} \newcommand{\vPi}{\boldsymbol{\Pi}} \newcommand{\vpsi}{\boldsymbol{\psi}} \newcommand{\vPsi}{\boldsymbol{\Psi}} \newcommand{\vrho}{\boldsymbol{\rho}} \newcommand{\vtheta}{\boldsymbol{\theta}} \newcommand{\vthetat}{\boldsymbol{\tilde{\theta}}} \newcommand{\vTheta}{\boldsymbol{\Theta}} \newcommand{\vsigma}{\boldsymbol{\sigma}} \newcommand{\vSigma}{\boldsymbol{\Sigma}} \newcommand{\vSigmat}{\boldsymbol{\tilde{\Sigma}}} \newcommand{\vsigmoid}{\vsigma} \newcommand{\vtau}{\boldsymbol{\tau}} \newcommand{\vxi}{\boldsymbol{\xi}}\\ % Lower Roman (Vectors) \newcommand{\va}{\mathbf{a}} \newcommand{\vb}{\mathbf{b}} \newcommand{\vBt}{\mathbf{\tilde{B}}} \newcommand{\vc}{\mathbf{c}} \newcommand{\vct}{\mathbf{\tilde{c}}} \newcommand{\vd}{\mathbf{d}} \newcommand{\ve}{\mathbf{e}} \newcommand{\vf}{\mathbf{f}} \newcommand{\vg}{\mathbf{g}} \newcommand{\vh}{\mathbf{h}} %\newcommand{\myvh}{\mathbf{h}} \newcommand{\vi}{\mathbf{i}} \newcommand{\vj}{\mathbf{j}} \newcommand{\vk}{\mathbf{k}} \newcommand{\vl}{\mathbf{l}} \newcommand{\vm}{\mathbf{m}} \newcommand{\vn}{\mathbf{n}} \newcommand{\vo}{\mathbf{o}} \newcommand{\vp}{\mathbf{p}} \newcommand{\vq}{\mathbf{q}} \newcommand{\vr}{\mathbf{r}} \newcommand{\vs}{\mathbf{s}} \newcommand{\vt}{\mathbf{t}} \newcommand{\vu}{\mathbf{u}} \newcommand{\vv}{\mathbf{v}} \newcommand{\vw}{\mathbf{w}} \newcommand{\vws}{\vw_s} \newcommand{\vwt}{\mathbf{\tilde{w}}} \newcommand{\vWt}{\mathbf{\tilde{W}}} \newcommand{\vwh}{\hat{\vw}} \newcommand{\vx}{\mathbf{x}} %\newcommand{\vx}{\mathbf{x}} \newcommand{\vxt}{\mathbf{\tilde{x}}} \newcommand{\vy}{\mathbf{y}} \newcommand{\vyt}{\mathbf{\tilde{y}}} \newcommand{\vz}{\mathbf{z}} %\newcommand{\vzt}{\mathbf{\tilde{z}}}\\ % Upper Roman (Matrices) \newcommand{\vA}{\mathbf{A}} \newcommand{\vB}{\mathbf{B}} \newcommand{\vC}{\mathbf{C}} \newcommand{\vD}{\mathbf{D}} \newcommand{\vE}{\mathbf{E}} \newcommand{\vF}{\mathbf{F}} \newcommand{\vG}{\mathbf{G}} \newcommand{\vH}{\mathbf{H}} \newcommand{\vI}{\mathbf{I}} \newcommand{\vJ}{\mathbf{J}} \newcommand{\vK}{\mathbf{K}} \newcommand{\vL}{\mathbf{L}} \newcommand{\vM}{\mathbf{M}} \newcommand{\vMt}{\mathbf{\tilde{M}}} \newcommand{\vN}{\mathbf{N}} \newcommand{\vO}{\mathbf{O}} \newcommand{\vP}{\mathbf{P}} \newcommand{\vQ}{\mathbf{Q}} \newcommand{\vR}{\mathbf{R}} \newcommand{\vS}{\mathbf{S}} \newcommand{\vT}{\mathbf{T}} \newcommand{\vU}{\mathbf{U}} \newcommand{\vV}{\mathbf{V}} \newcommand{\vW}{\mathbf{W}} \newcommand{\vX}{\mathbf{X}} %\newcommand{\vXs}{\vX_{\vs}} \newcommand{\vXs}{\vX_{s}} \newcommand{\vXt}{\mathbf{\tilde{X}}} \newcommand{\vY}{\mathbf{Y}} \newcommand{\vZ}{\mathbf{Z}} \newcommand{\vZt}{\mathbf{\tilde{Z}}} \newcommand{\vzt}{\mathbf{\tilde{z}}}\\ %%%% \newcommand{\hidden}{\vz} \newcommand{\obs}{\vy} \newcommand{\inputs}{\vu} \newcommand{\input}{\inputs}\\\newcommand{\hmmTrans}{\vA} \newcommand{\hmmObs}{\vB} \newcommand{\hmmInit}{\vpi} \newcommand{\hmmhid}{\hidden} \newcommand{\hmmobs}{\obs}\\\newcommand{\ldsDyn}{\vA} \newcommand{\ldsObs}{\vC} \newcommand{\ldsDynIn}{\vB} \newcommand{\ldsObsIn}{\vD} \newcommand{\ldsDynNoise}{\vQ} \newcommand{\ldsObsNoise}{\vR}\\\newcommand{\ssmDyn}{f} \newcommand{\ssmObs}{h}\end{aligned}\end{align} \]

# What are State Space Models?

A state space model or SSM is a partially observed Markov model, in which the hidden state, \(\hidden_t\), evolves over time according to a Markov process, possibly conditional on external inputs or controls \(\input_t\), and each hidden state generates some observations \(\obs_t\) at each time step. (In this book, we mostly focus on discrete time systems, although we consider the continuous-time case in XXX.) We get to see the observations, but not the hidden state. Our main goal is to infer the hidden state given the observations. However, we can also use the model to predict future observations, by first predicting future hidden states, and then predicting what observations they might generate. By using a hidden state \(\hidden_t\) to represent the past observations, \(\obs_{1:t-1}\), the model can have ``infinite’’ memory, unlike a standard Markov model.

Formally we can define an SSM as the following joint distribution:

(2)\[p(\hmmobs_{1:T},\hmmhid_{1:T}|\inputs_{1:T}) = \left[ p(\hmmhid_1|\inputs_1) \prod_{t=2}^{T} p(\hmmhid_t|\hmmhid_{t-1},\inputs_t) \right] \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1}) \right]\]

where \(p(\hmmhid_t|\hmmhid_{t-1},\inputs_t)\) is the transition model, \(p(\hmmobs_t|\hmmhid_t, \inputs_t, \hmmobs_{t-1})\) is the observation model, and \(\inputs_{t}\) is an optional input or action. See Figure 3 for an illustration of the corresponding graphical model.

Fig. 3 Illustration of an SSM as a graphical model.

We often consider a simpler setting in which there are no external inputs, and the observations are conditionally independent of each other (rather than having Markovian dependencies) given the hidden state. In this case the joint simplifies to

(3)\[p(\hmmobs_{1:T},\hmmhid_{1:T}) = \left[ p(\hmmhid_1) \prod_{t=2}^{T} p(\hmmhid_t|\hmmhid_{t-1}) \right] \left[ \prod_{t=1}^T p(\hmmobs_t|\hmmhid_t \right]\]

See Figure 4 for an illustration of the corresponding graphical model. Compare (2) and (3).

Fig. 4 Illustration of a simplified SSM.

# Hidden Markov Models

In this section, we discuss the hidden Markov model or HMM, which is a state space model in which the hidden states are discrete, so \(\hmmhid_t \in \{1,\ldots, K\}\). The observations may be discrete, \(\hmmobs_t \in \{1,\ldots, C\}\), or continuous, \(\hmmobs_t \in \real^D\), or some combination, as we illustrate below. More details can be found in e.g., [CMR05, Fra08, Rab89]. For an interactive introduction, see https://nipunbatra.github.io/hmm/.

## Example: Casino HMM

To illustrate HMMs with categorical observation model, we consider the “Ocassionally dishonest casino” model from [DEKM98]. There are 2 hidden states, representing whether the dice being used in the casino is fair or loaded. Each state defines a distribution over the 6 possible observations.

The transition model is denoted by

\[p(z_t=j|z_{t-1}=i) = \hmmTrans_{ij}\]

Here the \(i\)’th row of \(\vA\) corresponds to the outgoing distribution from state \(i\). This is a row stochastic matrix, meaning each row sums to one. We can visualize the non-zero entries in the transition matrix by creating a state transition diagram, as shown in Illustration of the casino HMM..

Fig. 5 Illustration of the casino HMM.

The observation model \(p(\obs_t|\hiddden_t=j)\) has the form

\[p(\obs_t=k|\hidden_t=j) = \hmmObs_{jk} \]

This is represented by the histograms associated with each state in Illustration of the casino HMM..

Finally, the initial state distribution is denoted by

\[p(z_1=j) = \hmmInit_j\]

Collectively we denote all the parameters by \(\vtheta=(\hmmTrans, \hmmObs, \hmmInit)\).

Now let us implement this model code.

    # state transition matrix
    A = np.array([
        [0.95, 0.05],
        [0.10, 0.90]
    ])

    # observation matrix
    B = np.array([
        [1/6, 1/6, 1/6, 1/6, 1/6, 1/6], # fair die
        [1/10, 1/10, 1/10, 1/10, 1/10, 5/10] # loaded die
    ])

    pi = np.array([0.5, 0.5])

    (nstates, nobs) = np.shape(B)

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